Eigenvalue problems on exterior domains and Dirichlet to Neumann maps

摘要

We consider a Schroedinger equation on an exterior domain in the case where the potential, which may be complex valued, has a limit at infinity. Associated with the problem is a Dirichlet to Neumann map on the inner boundary. We examine approximations to the problem obtained by (a) truncating the domain, (b) replacing the potential by its value at infinity outside some set, (c) a combination of (a) and (b). These approximations give rise to approximate Dirichlet to Neumann maps on the inner boundary. We analyze the convergence of these maps and deduce results on the approximation of the spectrum of the original problem by the spectra of the approximating problems. We show that, in theory, spurious eigenvalues cannot be generated by these procedures. Numerical experiments show, however, that the inherent ill-conditioning of a problem may cause spurious eigenvalues to appear due to the discretization of the truncated problems, even when the potential decays exponentially fast.